# Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction

arXiv: https://arxiv.org/abs/2312.08604 · HTML: https://mlsanity.com/paper/2312.08604/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 31 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Builds on (most cited first, 10 of 10)

- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Funnel Libraries for Real-Time Robust Feedback Motion Planning](https://mlsanity.com/paper/1601.04037/) (2016)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding](https://mlsanity.com/paper/2104.03270/) (2021)
- [Generating Formal Safety Assurances for High-Dimensional Reachability](https://mlsanity.com/paper/2209.12336/) (2022)
- [A Classification-based Approach for Approximate Reachability](https://mlsanity.com/paper/1803.03237/) (2018)
- [Technical Report: Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures](https://mlsanity.com/paper/1305.7484/) (2013)
- [Numerical Verification of Affine Systems with up to a Billion Dimensions](https://mlsanity.com/paper/1804.01583/) (2018)
- [Synthesis of separable controlled invariant sets for modular local control design](https://mlsanity.com/paper/1510.02990/) (2015)

## Built on by (most cited first, 23 of 23)

- [A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems](https://mlsanity.com/paper/2502.11057/) (2025)
- [Conformalized Interactive Imitation Learning: Handling Expert Shift and Intermittent Feedback](https://mlsanity.com/paper/2410.08852/) (2024)
- [One Filter to Deploy Them All: Robust Safety for Quadrupedal Navigation in Unknown Environments](https://mlsanity.com/paper/2412.09989/) (2024)
- [Conformal Predictive Programming for Chance Constrained Optimization](https://mlsanity.com/paper/2402.07407/) (2024)
- [Conformalized Teleoperation: Confidently Mapping Human Inputs to High-Dimensional Robot Actions](https://mlsanity.com/paper/2406.07767/) (2024)
- [Bridging conformal prediction and scenario optimization](https://mlsanity.com/paper/2503.23561/) (2025)
- [Synthesis and Deployment of Maximal Robust Control Barrier Functions through Adversarial Reinforcement Learning](https://mlsanity.com/paper/2604.13192/) (2026)
- [Statistically Assuring Safety of Control Systems using Ensembles of Safety Filters and Conformal Prediction](https://mlsanity.com/paper/2511.07899/) (2025)
- [Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions](https://mlsanity.com/paper/2505.11755/) (2025)
- [Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets](https://mlsanity.com/paper/2503.23912/) (2025)
- [Zono-Conformal Prediction: Zonotope-Based Uncertainty Quantification for Regression and Classification Tasks](https://mlsanity.com/paper/2508.11025/) (2025)
- [GenCtrl -- A Formal Controllability Toolkit for Generative Models](https://mlsanity.com/paper/2601.05637/) (2026)
- [Safe Control using Learned Safety Filters and Adaptive Conformal Inference](https://mlsanity.com/paper/2604.18482/) (2026)
- [From Refusal to Recovery: A Control-Theoretic Approach to Generative AI Guardrails](https://mlsanity.com/paper/2510.13727/) (2025)
- [Permissive Safety Through Trusted Inference: Verifiable Belief-Space Neural Safety Filters for Assured Interactive Robotics](https://mlsanity.com/paper/2606.02562/) (2026)
- [Robots that Suggest Safe Alternatives](https://mlsanity.com/paper/2409.09883/) (2024)
- [Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2410.02904/) (2024)
- [On Training-Conditional Conformal Prediction and Binomial Proportion Confidence Intervals](https://mlsanity.com/paper/2502.07497/) (2025)
- [MAD-PINN: A Decentralized Physics-Informed Machine Learning Framework for Safe and Optimal Multi-Agent Control](https://mlsanity.com/paper/2509.23960/) (2025)
- [Data-driven Reachable Set Estimation with Tunable Adversarial and Wasserstein Distributional Guarantees](https://mlsanity.com/paper/2604.12654/) (2026)
- [Multi-Variable Conformal Prediction: Optimizing Prediction Sets without Data Splitting](https://mlsanity.com/paper/2605.12341/) (2026)
- [VertexCBF: Improving Neural Control Barrier Functions via Vertex-Restricted Control Search](https://mlsanity.com/paper/2609.12831/) (2026)
- [CrossSafe: Towards Cross-Embodiment Latent Safety Filters](https://mlsanity.com/paper/2609.28984/) (2026)

## Abstract (arXiv)

Learning-based approaches for controlling safety-critical systems are rapidly growing in popularity; thus, it is important to assure their performance and safety. Hamilton-Jacobi (HJ) reachability analysis is a popular formal verification tool for providing such guarantees, since it can handle general nonlinear system dynamics, bounded adversarial system disturbances, and state and input constraints. However, its computational and memory complexity scales exponentially with the state dimension, making it intractable for large-scale systems. To overcome this challenge, neural approaches, such as DeepReach, have been used to synthesize reachable tubes and safety controllers for high-dimensional systems. However, verifying these neural reachable tubes remains challenging. In this work, we propose two verification methods, based on robust scenario optimization and conformal prediction, to provide probabilistic safety guarantees for neural reachable tubes. Our methods allow a direct trade-off between resilience to outlier errors in the neural tube, which are inevitable in a learning-based approach, and the strength of the probabilistic safety guarantee. Furthermore, we show that split conformal prediction, a widely used method in the machine learning community for uncertainty quantification, reduces to a scenario-based approach, making the two methods equivalent not only for verification of neural reachable tubes but also more generally. To our knowledge, our proof is the first in the literature to show a strong relationship between conformal prediction and scenario optimization. Finally, we propose an outlier-adjusted verification approach that uses the error distribution in neural reachable tubes to recover greater safe volumes. We demonstrate the efficacy of the proposed approaches for the high-dimensional problems of multi-vehicle collision avoidance and rocket landing with no-go zones.